Question

(a) Prove or disprove: if$ab\equiv 0\left(modn\right)$ , then$a\equiv 0\left(modn\right)$ or$a\equiv 0\left(modn\right)$ .

(b) Do part (a) when is prime.

Short answer

1. If$ab\equiv 0\left(modn\right)$ , neither$a\equiv 0\left(modn\right)$ or$b\equiv 0\left(modn\right)$ .
2. It is proved that $ab\equiv 0\left(modn\right)$;then$a\equiv 0\left(modn\right)$ or$b\equiv 0\left(modn\right)$ when$n$ is prime.

Step-by-step solution

Prove or disprove that if ab≡0 mod n , then a≡0 mod n  or b≡0 mod n 

It is given that$ab\equiv 0\left(modn\right)$ to prove or disprove $a\equiv 0\left(modn\right)$ or$b\equiv 0\left(modn\right)$ .

Consider , $n=6$, $a=2$and $b=3$, this implies that:

$2·3=0\left(mod6\right)$

But we can observe that role="math" localid="1646133742237" $2\not\equiv 0\left(mood6\right)$and$3\not\equiv 0\left(mod6\right)$ .

Hence, if $ab\equiv 0\left(modn\right)$, then neither $a\equiv 0\left(modn\right)$or$b\equiv 0\left(modn\right)$ .

Prove or disprove that if  ab≡0 mod n, then a≡0 mod n , or b≡0 mod n when  nis prime 

It is given that$ab\equiv 0\left(modn\right)$ to prove or disprove$a\equiv 0\left(modn\right)$ or$b\equiv 0\left(modn\right)$ when $n$is prime.

Since$n$ is prime and$ab\equiv 0\left(modn\right)$ , this implies that$\overline{)n}\left(ab\right)$

.According to the theorem, an integer $p$is prime if and only if $p$has the property whenever $\overline{)p}bc$, then$\overline{)p}b$ or$\overline{)p}c$ .

Usingthetheorem,$\overline{)n}a$ or $\overline{)n}b$this means that$a\equiv 0\left(modn\right)$ or$b\equiv 0\left(modn\right)$ .

Hence, it is proved that$ab\equiv 0\left(modn\right)$ ; then$a\equiv 0\left(modn\right)$ or$b\equiv 0\left(modn\right)$ when$n$ is prime.